1. Field of Invention
The invention relates to digital communications channel identification and equalization using measured second-order statistics of channel inputs when certain second-order statistics of channel inputs have periodically varying phase.
2. Description of Prior Art
In digital communications, the distortions caused by the physical channel between a transmitter and a receiver are incorporated into mathematical channel model. Possible distortions may include the well-known phenomenon of linear inter-symbol interference as well as additive noise and nonlinearity. Channel identification is the process of determining numeric values for the parameters of a channel model. These values allow the receiver to compensate for the channel distortions, a process known as equalization.
For general discussion of digital communications and equalization, see “Digital Communications” by J. G. Proakis (third edition, McGraw-Hill, 1995) and “Telecommunications Transmission Handbook” by R. L. Freeman (fourth edition, John Wiley & Sons, 1998).
Channel identification and equalization may rely on sending one or more known training symbols through the channel from the transmitter to the receiver. This training mode is used, for instance, in setting up a telephone modem connection. The modems at either end of the channel take a few seconds at the beginning of the connection to send training symbols and estimate equalizer parameters. The training symbols may be re-sent if one of the modems determines that the channel parameters have changed and that the parameter estimates are no longer accurate.
In another example of training-mode equalization, a digital mobile phone system using a standard such as GSM, EDGE, or TDMA sends bursts of data in packets. Training symbols occupy the middle of each packet. Because the channel parameters change from packet to packet, the receiver uses the training symbols to estimate the parameters for each packet separately.
Recent papers on equalization of digital mobile phone channels include “EDGE: Enhanced Data Rates for GSM and TDMA/136 Evolution” by A. Furuskar, S. Mazur, F. Muller, and H. Olofsson (IEEE Personal Communications, Vol. 6, No. 3, pp. 56-66, June 1999) and “Is Blind Channel Estimation Feasible in Mobile Communication Systems? A Study Based on GSM” by D. Boss, K.-D. Kammeyer, and T. Petermann (IEEE Journal on Selected Areas in Communications, Vol. 16, No. 8, pp. 1479-1492, October 1998).
The main advantages of training-based channel identification and equalization are that techniques for estimating the channel parameters can have low computational complexity and can produce parameter estimates relatively quickly. The main disadvantage of training-based channel identification and equalization is that some symbols that could be used to send data must instead be used for training symbols.
In some circumstances, few or no training symbols are available. A receiver may then rely on statistical techniques for channel parameter estimation. These techniques use knowledge of channel input statistics and measurements of channel output statistics. When there are no training symbols, channel identification and equalization are said to be “blind”. When there are a few training symbols, channel identification and equalization are said to be “semiblind”.
The main advantage of blind channel identification over training-based channel identification is that the transmitter does not have to reduce the data rate. The main drawbacks of blind channel identification are that the statistical signal processing techniques used may have a very high computational complexity and may not guarantee correct identification of every channel in a timely manner.
There are a wide variety of blind channel identification and equalization algorithms for channels modeled as having linear inter-symbol interference followed by independent additive noise. Blind methods for these linear channels may be broadly characterized as those based on higher-order statistics (HOS) and those based on second-order statistics (SOS).
The higher-order statistics in an HOS blind technique can be either explicit, such as in the Tricepstrum Equalization Algorithm, or implicit, such as in the Sato Algorithm and the Godard or Constant-Modulus Algorithm. HOS techniques with low computational complexity tend to have poor convergence properties. HOS techniques with good convergence properties tend to have very high computational complexity. Many HOS techniques are slow to converge and cannot identify all possible practical channels.
The Tricepstrum Equalization Algorithm is discussed in “Blind Equalization Based on Higher-Order Statistics (H.O.S.)” by D. Hatzinakos and C. L. Nikias (in “Blind Deconvolution”, S. Haykin editor, PTR Prentice Hall, 1994). The Sato Algorithm is discussed in “A Method of Recovering Equalization for Multilevel Amplitude-Modulation Systems” by Y. Sato (IEEE Transactions on Communications, Vol. COM-23, pp. 679-682, June 1975).
The Godard algorithm, which is also known as the Constant Modulus Algorithm, appears in “Self-Recovering Equalization and Carrier Tracking in Two-Dimensional Data Communication Systems” by D. N. Godard (IEEE Transactions on Communications, Vol. COM-28, pp. 1867-1875, November 1980) and in “A New Approach to Multipath Correction of Constant-Modulus Signals” by J. R. Treichler and B. G. Agee (IEEE Trans. on Acoustics, Speech, and Signal Processing, Vol. 31, No. 2, pp. 459-471, April 1983).
SOS techniques include those based on fractional sampling and those based on baud-sampled cyclostationary statistics.
Fractional sampling involves generating more than one channel output measurement for each transmitted symbol. For these fractional samples to provide useful information, the system must use extra transmission bandwidth.
Prior art discussions of fractional sampling for blind equalization appear in “Blind Identification and Equalization Based on Second-Order Statistics: A Time Domain Approach” by L. Tong, G. Xu, and T. Kailath (IEEE Transactions on Information Theory, Vol. 40, No. 2, pp. 340-349, March 1994) and in “Fractionally Spaced Equalization of Linear Polyphase Channels and Related Blind Techniques based on Multichannel Linear Prediction” by C. B. Papadias and D. T. M. Slock (IEEE Transactions on Signal Processing, Vol. 47, No. 3, pp. 641-654, March 1999) among others.
Advantages of SOS techniques based on fractional sampling include relatively low-complexity processing and fast convergence. Disadvantages are the inability to identify all practical linear channels, the need for extra transmission bandwidth, and processing at a sampling rate greater than the baud rate.
Baud-rate sampling involves generating one channel output measurement for each transmitted symbol. Because stationary baud-sampled SOS do not allow blind identification of channels not known either to be minimum-phase or to be maximum phase, the statistics must be cyclostationary. A cyclostationary statistic is one that changes in a periodic manner. The book “Cyclostationarity in Communications and Signal Processing” edited by W. A. Gardner (IEEE Press, 1994) has an extensive discussion of cyclostationary statistics.
Recent work on baud-sampled cyclostationary second-order statistics for blind equalization appears in the papers “Blind Channel Identification and Equalization with Modulation-Induced Cyclostationarity” by E. Serpedin and G. B. Giannakis (IEEE Transactions on Signal Processing, Vol. 46, No. 7, pp. 1930-1944, July 1998), “Filterbanks for Blind Channel Identification and Equalization” by G. B. Giannakis (IEEE Signal Processing Letters, Vol. 4, No. 6, pp. 184-187, June 1997), and “Transmitter Induced Cyclostationarity for Blind Channel Equalization” by M. K. Tsatsanis (IEEE Transactions on Signal Processing, Vol. 45, No. 7, pp. 1785-1794, July 1997).
All three of these papers focus on creating sequences with cyclostationary conjugated second-order moments. U.S. Pat. No 4,922,506 issued to R. D. McCallister and D. D. Shearer in 1990 proposes very broadly the similar idea of using cyclostationary properties of the cross-spectrum, which in sampled form is identical to use of conjugated second-order moments.
Advantages of SOS techniques based on baud-sampled cyclostationary statistics include low-complexity processing at the baud rate, fast convergence, and the ability to identify all practical linear channels. Disadvantages of many SOS techniques based on baud-sampled cyclostationary statistics include the need for extra transmitted power or extra transmitted bandwidth to produce the required cyclostationary properties.
For channels with nonlinearity, linear inter-symbol interference, and noise there are relatively few blind channel identification and equalization techniques. Most of these are based on higher-order statistics, and suffer from the disadvantages of HOS. HOS methods for nonlinear channels typically have much higher computational complexity than HOS methods for linear channels, whether blind or training-based.
Examples of nonlinear channel equalization appear in “Schemes for Equalisation of Communication Channels with Nonlinear Impairments” by S. Theodoridis, C. F. N. Cowan, C. P. Callender, and C. M. S. See (IEE Proceedings on Communications, Vol. 142, No. 3, pp. 165-171, June 1995) and in “A Novel Low Complexity Technique to Reduce Non-linear Distortion Effects in OFDM Systems” by D. Dardari, V. Tralli, and A. Vaccari (Proceedings of IEEE PIMRC '98, Boston, September 1998). An example of blind equalization using a nonlinear equalizer appears in “Using an RBF Network for Blind Equalization: Design and Performance Evaluation” by J. Gomes and V. Barroso (Proceedings of ICASSP 1997, pp. 3285-3288, 1997).
A common source of nonlinearity is a power amplifier at the transmitter that has a nonlinear transfer function. One example is in the paper by Dardari, Tralli, and Vaccari. Another appears in “Frequency Independent and Frequency Dependent Nonlinear Models of TWT Amplifiers” by A. M. Saleh (IEEE Transactions on Communications, Vol. COM-29, pp. 1715-1720, November 1981). For small input amplitudes, the output is highly linear, while for large input amplitudes, the output may be highly nonlinear. The nonlinearity may affect the amplitude, the phase, or both the amplitude and phase of the transmitted symbol.
One way to avoid the nonlinearity is to restrict the transmitter to using small input levels, a technique known as “back-off”. A disadvantage of back-off is that the power amplifier may then operate at a lower level of efficiency, with less useful, transmitted power relative to the power dissipated as heat. This can be a particular problem when the power source is limited, such as a mobile phone battery or a satellite solar panel.
Another way to combat the effects of certain types of power-amplifier nonlinearity is to use a symbol constellation having symbols that are all the same amplitude. If the nonlinearity is a function of the input amplitude, and all the symbols have the same amplitude, then the overall effect of the nonlinearity is a linear gain applied to the transmitted symbols. Using such a constant-modulus constellation, however, may limit the bandwidth efficiency. Bandwidth efficiency measures the number of data bits transmitted per symbol when transmitted power is limited. A constant modulus constellation may have symbols that are closer together than a non-constant modulus constellation with the same power and number of symbols. Fewer symbols must be used in order to meet minimum symbol separation requirements.
A third solution to channel identification and equalization of nonlinear channels is to use training symbols. This eases the difficulties of computational complexity and convergence, but at the cost of reduced data rates.
The disadvantages of existing methods for channel identification and equalization are several:                (a) Training-based channel identification and equalization techniques require that data rates be decreased in order to accommodate transmission of known training symbols.        (b) For linear channels, blind channel identification techniques based on higher-order statistics may have high computational complexity. They may also be slow to converge and may not be able to identify all commonly-occurring channels.        (c) For linear channels, blind channel identification techniques based on second-order statistics using fractional sampling require extra signal bandwidth, and cannot identify all commonly-occurring channels.        (d) For linear channels, blind channel identification techniques based on baud-sampled second-order statistics that are cyclostationary in amplitude require extra average transmitted power or extra signal bandwidth to provide useful cyclostationary properties.        (e) For nonlinear channels, blind channel identification algorithms based on higher-order statistics may have very high computational complexity. They may be slow to converge and may not be able to identify all commonly-occurring channels.        (f) For nonlinear channels, methods of reducing the effects of nonlinear distortion such as back-off or constant-modulus constellations can have undesirable effects on power efficiency or bandwidth efficiency.        